This paper presents a novel design of a quadruped hybrid leg-wheel mobile robot comprising two DOFs (a prismatic joint and a revolute joint) for the legs and one rotational joint for the wheel. The study begins by detailing the design of the structure and mechanism of the proposed system. We created the dynamic model of the proposed robot using MATLAB Simscape Multibody. The legged mode gait is developed using three supporting legs and one transferring leg. We developed a walking algorithm for the quadruped robot that comprises six motion steps. The simulation and experimental results demonstrate that our proposed robot has the ability to successfully traverse both flat surfaces and rough terrain. The robot’s design also enables it to perform various modes of movement, including normal vertical and horizontal driving, rotation around the robot’s centre of gravity, rotation in a large arc, and walking on flat terrain.